Finite Math Examples

$6 (16 x^{2} - 40 x + 25) + 3$

Step 1

Set $6 (16 x^{2} - 40 x + 25) + 3$ equal to $0$ .

$6 (16 x^{2} - 40 x + 25) + 3 = 0$

Step 2

Simplify the left side.

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Step 2.1

Simplify $6 (16 x^{2} - 40 x + 25) + 3$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Apply the distributive property.

$6 (16 x^{2}) + 6 (- 40 x) + 6 \cdot 25 + 3 = 0$

Step 2.1.1.2

Simplify.

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Step 2.1.1.2.1

Multiply $16$ by $6$ .

$96 x^{2} + 6 (- 40 x) + 6 \cdot 25 + 3 = 0$

Step 2.1.1.2.2

Multiply $- 40$ by $6$ .

$96 x^{2} - 240 x + 6 \cdot 25 + 3 = 0$

Step 2.1.1.2.3

Multiply $6$ by $25$ .

$96 x^{2} - 240 x + 150 + 3 = 0$

Step 2.1.2

Add $150$ and $3$ .

$96 x^{2} - 240 x + 153 = 0$

Step 3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4

Substitute the values $a = 96$ , $b = - 240$ , and $c = 153$ into the quadratic formula and solve for $x$ .

$\frac{240 \pm \sqrt{{(- 240)}^{2} - 4 \cdot (96 \cdot 153)}}{2 \cdot 96}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 240$ to the power of $2$ .

$x = \frac{240 \pm \sqrt{57600 - 4 \cdot 96 \cdot 153}}{2 \cdot 96}$

Step 5.1.2

Multiply $- 4 \cdot 96 \cdot 153$ .

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Step 5.1.2.1

Multiply $- 4$ by $96$ .

$x = \frac{240 \pm \sqrt{57600 - 384 \cdot 153}}{2 \cdot 96}$

Step 5.1.2.2

Multiply $- 384$ by $153$ .

$x = \frac{240 \pm \sqrt{57600 - 58752}}{2 \cdot 96}$

Step 5.1.3

Subtract $58752$ from $57600$ .

$x = \frac{240 \pm \sqrt{- 1152}}{2 \cdot 96}$

Step 5.1.4

Rewrite $- 1152$ as $- 1 (1152)$ .

$x = \frac{240 \pm \sqrt{- 1 \cdot 1152}}{2 \cdot 96}$

Step 5.1.5

Rewrite $\sqrt{- 1 (1152)}$ as $\sqrt{- 1} \cdot \sqrt{1152}$ .

$x = \frac{240 \pm \sqrt{- 1} \cdot \sqrt{1152}}{2 \cdot 96}$

Step 5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{240 \pm i \cdot \sqrt{1152}}{2 \cdot 96}$

Step 5.1.7

Rewrite $1152$ as $24^{2} \cdot 2$ .

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Step 5.1.7.1

Factor $576$ out of $1152$ .

$x = \frac{240 \pm i \cdot \sqrt{576 (2)}}{2 \cdot 96}$

Step 5.1.7.2

Rewrite $576$ as $24^{2}$ .

$x = \frac{240 \pm i \cdot \sqrt{24^{2} \cdot 2}}{2 \cdot 96}$

Step 5.1.8

Pull terms out from under the radical.

$x = \frac{240 \pm i \cdot (24 \sqrt{2})}{2 \cdot 96}$

Step 5.1.9

Move $24$ to the left of $i$ .

$x = \frac{240 \pm 24 i \sqrt{2}}{2 \cdot 96}$

Step 5.2

Multiply $2$ by $96$ .

$x = \frac{240 \pm 24 i \sqrt{2}}{192}$

Step 5.3

Simplify $\frac{240 \pm 24 i \sqrt{2}}{192}$ .

$x = \frac{10 \pm i \sqrt{2}}{8}$